The Republican primary for a Senate seat in North Carolina Tuesday offers the most desperate political prognosticators a chance to read the tobacco leaves heading into 2016.

That’s because former Florida Gov. Jeb Bush, Kentucky Sen. Rand Paul and former Arkansas Gov. Mike Huckabee – three men eying White House bids in 2016 – have all thrown their support behind separate candidates.

Last week, Mr. Bush endorsed Thom Tillis, the speaker of the North Carolina House who is backed by GOP leaders in Washington and a pair of prominent outside groups, American Crossroads and the U.S. Chamber of Commerce. Mr. Paul, meanwhile, is supporting obstetrician Greg Brannon, and Mr. Huckabee blessed Baptist pastor Mark Harris.

All three potential White House hopefuls represent a disparate wing of the GOP, and so do the candidates they’re supporting: Mr. Tillis is the “establishment” pick; Mr. Brannon is the favorite of libertarians and tea-party activists; and Mr. Harris is the preferred pick of many evangelicals, having led a successful campaign to amend North Carolina’s constitution to define marriage as an institution between one man and one woman.

Tuesday’s North Carolina primary will give handicappers the first chance to gauge which faction has the upper-hand heading into a series of contentious nominating contests later this year, as well as 2016 when the party will determine its long-term course by picking a White House nominee.

A poll released Monday shows Mr. Tillis right on the cusp of the 40% of the primary vote he needs to avoid a costly runoff that won’t be resolved until mid-July. Mr. Tillis has built a solid lead, according to the latest numbers from the Democratic firm Public Policy Polling, as the preferred pick of 40% of the more than 900 likely Republican primary voters surveyed.

Mr. Brannon was favored by 28% of the likely electorate, while 15% backed Mr. Harris. Eleven percent of those polled remain undecided in the race, giving Mr. Tillis an audience of potentially persuadable voters to push him over the 40% threshold. The automated telephone survey has a margin of error of 3.2 percentage points.

Mr. Paul was scheduled to appear in Charlotte with Mr. Brannon on Monday for a Grassroots Victory Rally at the NASCAR Hall of Fame.

Mr. Tillis got a fresh boost from 2012 GOP presidential nominee Mitt Romney, who endorsed the North Carolina House speaker on Monday.

Mr. Bush highlighted Mr. Tillis’s conservatism in his endorsement last week, calling the House speaker a “proven conservative with an impressive track record of results.” But the Florida governor also made the case that Mr. Tillis offers the GOP its best chance to unseat Democratic Sen. Kay Hagan, a top target of Senate Republicans and their outside allies.

“It is critically important that Republicans win a majority in the U.S. Senate, and I am confident that the road to a majority runs through Thom Tillis in North Carolina,” Mr. Bush said.

The quotes – and the rationale – could be applied to a number of other party-backed candidates in other states where there are competitive primaries, including other open primaries in Georgia, Iowa and Oregon and races where incumbent Republicans face a conservative challenge in Kansas, Kentucky and Mississippi, among others.

On Monday, Florida Sen. Marco Rubio, another potential White House contender in 2016, reversed his previous pledge not to get involved in competitive primaries by endorsing state Sen. Joni Ernst in Iowa’s crowded contest. The primary hasn’t drawn quite the attention that the North Carolina race has at this point, but it wouldn’t be all that surprising to see another proxy fight break out in the first-in-the-nation caucus state, where these early gestures tend to have ramifications later on.

These endorsements tend to fuel further bickering between the competing wings of the party.

Mr. Brannon responded to the Bush endorsement of Mr. Tillis last week by telling conservative radio host Glenn Beck that the nod is just another example of the so-called “establishment” trying to deny conservative activists their preferred pick.

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Washington Wire is one of the oldest standing features in American journalism. Since the Wire launched on Sept. 20, 1940, the Journal has offered readers an informal look at the capital. Now online, the Wire provides a succession of glimpses at what’s happening behind hot stories and warnings of what to watch for in the days ahead. The Wire is led by Reid J. Epstein, with contributions from the rest of the bureau. Washington Wire now also includes Think Tank, our home for outside analysis from policy and political thinkers.